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The Product Rule is a fundamental technique in calculus for finding derivatives of products of two differentiable functions. Let's say you have two functions, \( u(x) \) and \( v(x) \), and you want to differentiate their product \( u(x) \cdot v(x) \) with respect to \( x \). The Product Rule states that the derivative is given by: \[ (u \cdot v)' = u' \cdot v + u \cdot v' \] This means you take the derivative of the first function, multiply it by the second function as is, then add the first function unchanged multiplied by the derivative of the second function.

• Assign \( u(x) = 1 + \sec \theta \) and \( v(x) = \sin \theta \)
• Differentiate each part: \(u' = \sec \theta \tan \theta\) and \(v' = \cos \theta\)
• Apply: \( (1 + \sec \theta) \cos \theta + (\sec \theta \tan \theta) \sin \theta \)

By applying the Product Rule correctly, you can find the derivative of more complex expressions, like the one in the exercise.

Trigonometric functions, such as sine, cosine, and secant, are utilized widely in calculus, especially when dealing with circular or periodic functions. These functions describe the relationships between the angles and sides of triangles. Here's a quick refresher on the key trigonometric functions involved in our exercise:

• \( \sin \theta \) is the y-coordinate of the point on the unit circle.
• \( \cos \theta \) is the x-coordinate of the same point.
• \( \sec \theta \) is defined as the reciprocal of \( \cos \theta \), hence \( \sec \theta = \frac{1}{\cos \theta} \).
• \( \tan \theta \) is the ratio of \( \sin \theta \) to \( \cos \theta \), so \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

These functions can be challenging, but once you understand the relationships between them, applying them in differentiation becomes straightforward. Remember, each function has its own derivative:

• \( \frac{d}{d\theta} (\sin \theta) = \cos \theta \)
• \( \frac{d}{d\theta} (\cos \theta) = -\sin \theta \)
• \( \frac{d}{d\theta} (\sec \theta) = \sec \theta \tan \theta \)

These are essential for performing the differentiation tasks in calculus exercises.

In this differentiation exercise, we have the function \( r = (1 + \sec \theta) \sin \theta \) with respect to the variable \( \theta \). The task is to find the derivative \( \frac{dr}{d\theta} \). To solve this, use the Product Rule since the function is a product of two parts, \( (1 + \sec \theta) \) and \( \sin \theta \). Start by differentiating each part separately. The main steps are:

• Differentiate \(1 + \sec \theta\) to get \( \sec \theta \tan \theta \).
• Differentiate \( \sin \theta \) to get \( \cos \theta \).
• Substitute these derivatives into the Product Rule formula.
• Simplify the expression using trigonometric identities.

For the final expression, you combine all terms and simplify:\[ \frac{dr}{d\theta} = \sec \theta \tan \theta \sin \theta + \cos \theta + \sec \theta \cos \theta \] With further simplification and substituting known identities for trigonometric functions, you will get a neat final form. Remembering these steps and understanding trigonometric relationships will help you repeat this process across any similar exercise.